I know the area of a sphere is 4phi(r^2), but I'm wondering how to
derive that formula. I know it should be done in cylindrical
coordinates, and I'm thinking that the arc of a circle is defined as
rd(theta) and it's multiplied with rd(phi) to get (r^2)d(theta)d(phi).
Could you please help explain this?

In the end elevation of a drawing, a rod rises from point A at 32
degrees. In the side elevation, the same rod is seen rising from point
A at 48 degrees. How do I work out at what angle to cut the end of the
rod?

Draw the locus of all points in the plane that are equidistant from
the rays of an angle and equidistant from two points on one side of
the angle. I'm having trouble with this kind of locus problem. Can you
explain how to think about and solve them?

Given two points in 3-D space, such as A(x1,y1,z1) and B(x2,y2,z2),
what would be the equation of the line that connects those points? I
know that in the 2-D plane the equation of a line in slope-intercept
form is y = mx + b. Is there something similar in 3-D?

I have a height map of a terrain where the x and y values are fixed. I
can calculate best-fit slopes in the x and y directions, but I can't
figure out how to combine them into a best-fit regression plane.

I have solar collectors on my roof. They are mounted so that the base
of each panel runs up the slope of the roof, and the panels themselves
are mounted at an angle. I'd like to know how to determine the various
angles created by this situation.